\chapter{椭球体 Ecllipsoid}
\section{为什么好多天体最终变成了球体？}
https://www.zhihu.com/question/22121767

旋转物体在引力作用下的能量最低状态是麦克劳林椭球。只要星体足够大，引力势能足够产生高温高压熔化星体，那么液态的星体自然就会变成椭球。这个椭球的形状可以由星体密度与旋转角速度唯一确定由方程\ref{OMSRRPSeq2}：

孔大力博士论文 ON THE GRAVITATIONAL FIELDS OF MACLAURIN SPHEROID MODELS OF ROTATING FLUID PLANETS

https://iopscience.iop.org/article/10.1088/0004-637X/764/1/67

Hubbard最近推导了一个重要的迭代方程，用于计算Maclaurin球体的重力系数，该常数不需要在小畸变参数中进行扩展。 我们表明，当失真参数不够小时，基于Poisson方程的不完全解的迭代方程会发散。 我们基于泊松方程的完整解导出一个新的迭代方程，因此，在计算麦克劳林椭球体的重力系数时，总会收敛。

The dawning of the theory of equilibrium figures: a brief historical account from the 17th through the 20th century
Giuseppe Iurato
(Submitted on 12 Sep 2014 (v1), last revised 8 Nov 2014 (this version, v2))
A brief but complete historical survey of the theory of equilibrium figures from its early origins, dating back to 17th-century, until the latest 20th-century developments, with a view towards its applications, is carried out. 

https://arxiv.org/abs/1409.3858v2

钱德拉萨卡

钱德拉塞卡 椭球平衡图：历史记述

Ellipsoidal figures of equilibrium—an historical account

S. Chandrasekhar 
First published: May 1967 https://doi.org/10.1002/cpa.3160200203 Citations: 31 

https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160200203

1895年 Dyson较严格地计算了子午面为正圆和椭圆的环状星体的引力场。  https://royalsocietypublishing.org/toc/rsta/184/1041

1977年，Marcus等人使用数值方法研究了轴对称旋转自引力流体的稳定性问题。https://ui.adsabs.harvard.edu/abs/1977ApJ...214..584M

Stablest Shapes for an Axisymmetric Body of Gravitating, Incompressible Fluid 
Marcus, Philip S. ; Press, William H. ; Teukolsky, Saul A. 
Abstract
A numerical method for computing the total energy of self-gravitating, incompressible, rotating, axisymmetric fluid bodies is presented. Then, using a minimization technique, the stablest axisymmetric shapes are found for fluids having the same angular momentum distribution as the Maclaurin spheroids. For small angular momenta the Maclaurin spheroid is a minimum-energy configuration; above a certain value a new, toroidal family of differentially rotating figures becomes the stable minimum-energy shape. Just below this critical value the spheroids are stable to small perturbations, but the corresponding toroids have lower energy. The family of "Mestel disks" (mass cc 1 /r, flat rotation curve) with this same angular momentum distribution are equilibria, but they are always unstable. Similar conclusions hold for other angular momentum distributions also investigated. These results may clarify the "ring formation" stage of some realistic collapse models, and may also support the hypothesis of massive galactic halos. Subject headings: galaxies: formation - hydrodynamics - instabilities - rotation 


Publication: 
The Astrophysical Journal
Pub Date: June 1977 DOI: 10.1086/155284  Bibcode: 1977ApJ...214..584M  



@ARTICLE{1977ApJ...214..584M,
	author = {{Marcus}, Philip S. and {Press}, William H. and {Teukolsky}, Saul A.},
	title = "{Stablest Shapes for an Axisymmetric Body of Gravitating, Incompressible Fluid}",
	journal = {apj},
	year = 1977,
	month = jun,
	volume = {214},
	pages = {584-597},
	doi = {10.1086/155284},
	adsurl = {https://ui.adsabs.harvard.edu/abs/1977ApJ...214..584M},
	adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

\section{Gaia}
http://www.esa.int/Science\_Exploration/Space\_Science/Gaia/The\_legend\_of\_Gaia
\subsection{about Gaia}
For millennia astronomers have looked to the sky and gazed in wonder at the stars and planets. Ancient civilisations already realised that objects in the sky appeared to move in a regular manner, and many communities used the stars to determine when to plant and harvest their crops.
Thus began the oldest branch of astronomy – astrometry – that is, the study of the geometrical relationship between objects in the sky and their apparent and true motions. For many centuries, astronomers have devoted their time to the art of determining the relative position of the stars, a fundamental requirement for cataloguing the night sky.
The true pioneer of the science of astrometry was the ancient Greek astronomer Hipparchus, who in 129 BC, and with only naked eye observations and simple geometry, catalogued the relative positions of around one thousand stars. He determined their relative brightness and positions with an accuracy of about one degree (the angle equivalent to the apparent height of a person at a distance of 100 m).
Progress in the accuracy of measuring angles only accelerated in the 16th century with astronomer Tycho Brahe’s stellar observations using sextants and quadrants. He managed to fix star positions to an accuracy of about half an arcminute (an arcminute is 1/60 of a degree).  
In 1609 the telescope was invented and later in the same century instruments that could be used with telescopes to determine angles in space. Finally, astronomers could determine angles to accuracies greater than the human eye can see.
By the 18th century, accuracies improved to the order of arcseconds, and by the 19th century, fractions of a second of arc.
This finally opened up the ability to measure stellar parallax, the change in the apparent position of a star when viewed from two widely separated positions, for example as viewed from Earth 6 months either side of its orbit around the Sun. Using trigonometry, the parallax angle can be converted to a precise distance. But it is an extremely small quantity – even the nearest star has a parallax of less than 1 arcsecond – making the measurement accessible only to the most sensitive instruments, and only for the nearest stars.
Moreover, these early parallax measurements were limited by viewing through Earth’s atmosphere, which distorted the results and put an upper limit on their accuracy. The only way to really achieve precise measurements would be from space.
In 1989, more than 2000 years after Hipparchus first looked at the heavens, ESA launched a mission named in his honour: theHighPrecisionParallaxCollectingSatellite. ESA’s Hipparcos was the first satellite ever devoted to astrometry, and revolutionised the field of precision astronomy, improving on accuracies achieved on the ground 100 times over, down to just 1 milliarcsecond.
Data were collected between 1989 and 1993, with the resulting Hipparcos Catalogue published in 1997. It contains the positions, distances and movements – 200 times more accurate than any previous measurement – for almost 120 000 stars. A second, larger, catalogue – the Tycho catalogue – contains data of 2.5 million stars to a lesser precision. These catalogues set the precedent on stellar positions and are continuously used in space science research and for spacecraft navigation.
Gaia will continue the noble European legacy of star charting. It is destined to catalogue a thousand million stars, measuring each star's position and motion 200 times more accurately than the Hipparcos mission, and producing 10 000 times more data than its predecessor. Equipped with information about each star’s position and velocity, astronomers will be able to trace the past trajectory of stars, thus ultimately deciphering the history of the Milky Way.
Gaia’s extraordinary precision represents the dream of astronomers throughout history, and will bring to light answers to the many questions that have been asked along the way.
\subsection{关于盖亚}
几千年来，天文学家一直仰望天空，凝视着恒星和行星。古代文明已经意识到，天空中的物体似乎有规律地运动，许多社区使用星星来确定何时种和收割庄稼。这样就开始了最古老的天文学分支-天文测量-即研究天空中的物体与其视在运动和真实运动之间的几何关系。许多世纪以来，天文学家一直致力于确定恒星相对位置的技术，这是对夜空进行分类的基本要求。天文学科学的真正开创者是古希腊的天文学家希帕科斯（Hipparchus），他在公元前129年仅凭肉眼观察和简单的几何结构就对大约1000颗恒星的相对位置进行了分类。他确定了它们的相对亮度和位置，精确度约为1度（该角度等于人在100 m处的视在高度）。天文学家第谷·布拉赫（Tycho Brahe）使用六分仪和象限进行的恒星观测，直到16世纪才加速了角度测量的准确性。他设法将恒星位置固定到大约半个弧度（一弧度为1/60度）。1609年发明了望远镜，在同一世纪末期，可以将其与望远镜一起使用来确定空间角度的仪器也已问世。最后，天文学家可以确定比人眼可以看到的更大的精度角度。到18世纪，精确度提高到了弧秒的数量级，到19世纪，精确度提高到了几分之一秒。最终，这开启了测量恒星视差的能力，恒星视差是从两个相距很远的位置观察到的，例如从地球绕太阳公转的六个月观察时，观察到的恒星表观位置发生了变化。使用三角学，可以将视差角转换为精确的距离。但是它的数量非常小-即使最近的恒星的视差也小于1弧秒-使得只有最敏感的仪器才能测量到该数据，并且只有最近的恒星才能进行测量。此外，这些早期的视差测量受到地球大气层观测的限制，从而扭曲了结果，并提高了其准确性的上限。真正实现精确测量的唯一方法是从太空。1989年，在希帕基斯首次注视天空2000多年后，欧空局发起了以他的名字命名的任务：高精度视差收集卫星。 ESA的Hipparcos是有史以来第一颗致力于天文测量的卫星，它彻底改变了精密天文学领域，将地面精度提高了100倍，降至仅1毫秒。收集了1989年至1993年的数据，最终的Hipparcos目录于1997年发布。它包含的位置，距离和运动–精确度是以前任何测量的200倍–接近12万颗恒星。第二个更大的目录-第谷目录-包含250万颗恒星的数据，但精度较低。这些目录树立了恒星位置的先例，并在航天科学研究和航天器导航中不断使用。盖亚（Gaia）将继续保留星图图表在欧洲的崇高传统。它注定要对十亿颗恒星进行分类，测量每个恒星的位置和运动要比Hipparcos任务精确200倍，并且产生的数据比其前任要多1万倍。有了有关每颗恒星的位置和速度的信息，天文学家将能够追踪过去的恒星轨迹，从而最终解释银河系的历史。盖亚的非凡精确性代表了整个历史上天文学家的梦想，并将揭露沿途许多问题的答案。
